March 14th is Pi(e) Day, and this year we get an extra two digits (this year’s date being, in the American style, 3/14/15, taking us through the first 5 digits of pi). While many people (including our culinarily-inclined staff here at the Folger) celebrate this day with sweet (and not-so-sweet) pastries, I wanted to bring the day back to its roots, and explore the mathematical side of Pi Day.

Symbolic algebra is something that most of us take for granted today; it’s how we are taught in school from very early on, and it becomes so ingrained in our language that the symbols (and their attached meanings) have started to move out of the realm of mathematics and into our everyday lives, as anyone who has ever used a character-limited media like Twitter knows well.

But where did this start? When did the symbol = come to mean “is equal to,” + mean “plus”, and when did the ratio of a circle’s circumference to its diameter get called by the Greek letter pi, and then when did it shift to the *actual* Greek letter π?

For the answer in English, at least, we have to begin with Robert Recorde. In between being a physician and the administrator of the Bristol mint, Recorde was a mathematician. His first publication, *The Grounde of Artes *(1543; STC 20797.5), was a basic introduction to arithmetic, written in dialog form, that was enormously popular and was reissued many times, for over a century after his death. ^{1}

This dialog format was not uncommon for mathematical treatises of the time, and the idea of writing out mathematical expressions in a symbolic (rather than prose) form had not yet taken hold in English—not, that is, until 1557, when Recorde decided to supplement *Grounde* with a more advanced treatise on arithmetic and algebra, which he titled *The Whetstone of Witte* (STC 20820).

This publication introduced three important mathematical symbols to the English-reading world. First, he borrowed from German mathematicians, and produced what seems to have been the first time the plus and minus signs were used in English:

Even more importantly, though, Recorde declared:

Yes, Recorde got sick of writing “is equal to” over and over again and invented the equal sign so he wouldn’t have to!The introduction of these three symbols allowed, for the first time in English, algebraic equations to be written.

Some 90 years later, the English translation of William Oughtred’s *Clavis mathematica* was published as *The Key of the Mathematics New Forged and Filed* (1647; 154- 247q), and the x as multiplication sign was introduced:

Thus, by the mid 17th century, nearly all of the components were in place in English to express mathematical equations symbolically rather than in prose. This was critical for mathematical and scientific expressions to grow in complexity and allow the likes of James Gregory and Isaac Newton to begin working out infinite series (that is, numbers that can be represented by an infinite sum of numbers such as ½ + ¼ + ⅛ + etc. ^{2} ).

So where does that leave π? ^{3}

The fact that the ratio of a circle’s diameter to its circumference is constant no matter the size of the circle has been known to civilizations around the world since antiquity. Knowledge of that particular ratio, 22:7, was not far behind. Pi had long been approximated, using various methods, and as our understanding of mathematics has grown in complexity, so too have the approximations.

For the most part, mathematicians would either use a ratio, or would pick a certain number of decimal places and use that as their approximation. For example, in Oughtred’s second book, *The Circles of Proportion and the Horizontal Instrument* (1632; STC 18899a), he (like his contemporaries) used the ratio and decimal representation of pi:

But the fact that this number was an approximation was always there, lurking around the edges of everyone’s calculations. Not to mention, as the approximations grew more and more complex, the harder it became to integrate into a text as an equation, let alone as prose.

The first time we see the Greek letter π used in connection with circles is in Oughtred’s 1647 *Key of the Mathematics*. Here, Oughtred used π to represent the periphery (or circumference) of a circle and ∂ to represent the diameter in the ratio.

The thought was certainly there, that it made sense to represent this constant symbolically. But Oughtred was just as clearly caught up by the fact that it was a ratio of two aspects, and so he chose to give symbols to each part, rather than to the constant as a whole.

It wasn’t until the beginning of the 18th century that π started to be used in its current way, thanks to William Jones’s *A New Introduction to the Mathematics* (1706; 167- 873q). ^{4}

In this book, a compilation of his teaching notes, Jones started at the very beginning, making sure his readers understood the mathematical symbols he was going to use.

When it came to the discussion of circles, and of pi, Jones started by showing the approximation of the ratio, out to one hundred places, crediting John Machin.

Then, in a later discussion of the uses of infinite series, he once again returns to the diameter to circumference ratio of a circle:

Here, at last, we have this ratio fully represented by π.

Unfortunately for Jones’s place in history, this use of π did not really catch on in the larger mathematical/scientific world until about 30 years later, when noted Swiss mathematician and physicist Leonhard Euler began using it in his texts.

But happily for us today, we now take as a given that π = 3.14, and that allows us to spend a whole day in March, each year, giving thanks to mathematicians like Recorde, Oughtred, and Jones.

Or by eating pastries. Whichever floats your boat.

- Stephen Johnston, ‘Recorde, Robert (c.1512–1558)’, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, Jan 2008 [http://www.oxforddnb.com/view/article/23241].
- Can you imagine trying to write that out in prose?
- Insert your own Life of Pi jokes here.
- Ruth Wallis, ‘Jones, William (c.1675–1749)’, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, Sept 2012 [http://www.oxforddnb.com/view/article/15104]

We can do this again next year, because 3.14159 rounds to 3.1416.

Interesting also that the Romans seemed quite happy to adopt and use the Greek word ‘diametros’ but balked at ‘perimetros’, preferring their own ‘circumferentia’ for the latter. So today we’re stuck with one Greek and one Latin word for the two aspects of π, even though we still use the Greek letter ‘π’ instead of ‘c’ for the circumference of a circle whose diameter is 1.

Thanks so much for this post, Abbie. It got me thinking about type supply and whether you noticed a correlation between printers who use π in mathematical treatises and those who were already printing books that required Greek type sorts.

The other thing that occurred to me is that original research about the history and use of special characters (like π, ¶, etc.) in printing is only possible through contact with physical books or the browsing of digital resources like EEBO since such digital corpora/repositories do not (yet!) permit us to search directly for these special characters. This said, all the more credit to you for your work here!

Fascinating blog entry – thanks. It would be useful to hear some further remarks on the “underground” status of math in early modern thinking and, more generally, how this sort of applied mathematics relates to the enormously influential but still poorly understood or appreciated neo-Pythagorean doctrines of the age.